Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, r\neq 0$. $\dfrac{{(p^{5}r^{5})^{-3}}}{{(p^{5}r)^{-3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{5}r^{5})^{-3} = (p^{5})^{-3}(r^{5})^{-3}}$ On the left, we have ${p^{5}}$ to the exponent ${-3}$ . Now ${5 \times -3 = -15}$ , so ${(p^{5})^{-3} = p^{-15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{5}r^{5})^{-3}}}{{(p^{5}r)^{-3}}} = \dfrac{{p^{-15}r^{-15}}}{{p^{-15}r^{-3}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-15}r^{-15}}}{{p^{-15}r^{-3}}} = \dfrac{{p^{-15}}}{{p^{-15}}} \cdot \dfrac{{r^{-15}}}{{r^{-3}}} = p^{{-15} - {(-15)}} \cdot r^{{-15} - {(-3)}} = r^{-12}$